Abstract

We examine the logarithmic corrections to the black hole (BH) entropy product formula of outer horizon and inner horizon by taking into account the effects of statistical quantum fluctuations around the thermal equilibrium and via conformal field theory (CFT). We argue that, in logarithmic corrections to the BH entropy product formula when calculated using CFT and taking into account the effects of quantum fluctuations around the thermal equilibrium, the formula should not be universal and it also should not be quantized. These results have been explicitly checked by giving several examples.

1. Introduction

There has been considerable ongoing excitement in physics of BH thermodynamic product formula [particularly area (or entropy) product formula] of inner horizon () and outer horizon () for a wide variety of BHs [1–9] which have been examined so far without considering any logarithmic correction. For several cases, the product is mass-independent (universal) and in some specific cases the product is not mass-independent. This investigation has been more sparked by Cvetič et al. [2] for supersymmetric BPS (Bogomol’ni-Prasad-Sommerfield) class of BHs which have inner and outer BH entropy of the form , where and are excitation numbers of the left and right moving sectors of a weakly coupled two-dimensional (2D) CFT. Therefore, their product   must be quantized in nature.

It has been suggested by Larsen [10] that BH event horizon is quantized in Planck units so it is natural to be valid for Cauchy horizon also. This also indicates that the product of area (or entropy) is quantized in terms of quantized charges and quantized angular momentum and so on. But this has been discussed without considering any logarithmic correction to the BH entropy. Now if we take into account the logarithmic correction to the BH entropy, what happens in case of logarithmic correction to the BH entropy product formula? What are the quantization rules for logarithmic corrected BH entropy product formula? These are the main issues that we will be discussed in this work.

It should be noted that this is the continuation of our earlier investigation [11]. In the previous work, we derived the general logarithmic correction to the entropy product formula of event horizon and Cauchy horizon for various spherically symmetric and axisymmetric BHs by taking into account the effects of quantum fluctuations around the thermal equilibrium. These corrections are evaluated in terms of some BH thermodynamic parameters, namely, the specific heat   of and BH temperature   of , respectively.

The logarithmic correction of BH entropy of is described by the key formulawhere is density of states of and is entropy of and their product is derived to bewhere is the entropy of without logarithmic correction.

In the present work, we shall compute the general logarithmic correction to BH entropy of whenever we have taken the effects of statistical quantum fluctuations around the thermal equilibrium and by using exact entropy function according to the formalism borrowed from the quantum theory of gravity [12–25]. Whenever we incorporated the effects of quantum fluctuations around the thermal equilibrium, the Bekenstein-Hawking entropy formula must be corrected and the entropy product formula of must also be corrected. This is the main motivation behind this work.

It has been known that BHs in Einstein’s gravity as well as other theories of gravity are much larger than the Planck scale length where the Bekenstein-Hawking entropy is precisely proportional to the horizon area [12, 13, 18–20]. Thus, it is quite natural to investigate the leading order corrections in Bekenstein-Hawking entropy as well as in Bekenstein-Hawking entropy product formula of and when one can reduce the size of BH. For large BHs, it has been proved that the logarithm of the density of states is exactly the Bekenstein-Hawking entropy plus the corrections term , where and is the area of the event horizon [22, 23]. Thus, it appears that the logarithmic corrections to the Bekenstein-Hawking entropy as well as Bekenstein-Hawking entropy product formula of are a generic feature of BHs. It has been verified earlier by Das et al. [24] for and here we have tried to examine for followed by our earlier investigation [11].

It should be emphasized that logarithmic corrections to the Bekenstein-Hawking formula are very interesting and a great deal about such corrections is known in string theory and beyond. Logarithmic corrections arise from various sources, the simplest of which are the statistical fluctuations around thermal equilibrium. These are always present because they arise from saddle point Gaussian corrections to the integral that computes the density of states from the partition function. In some cases, such as the BTZ BH in pure 3D gravity, these are the only logarithmic corrections to the Bekenstein-Hawking entropy. However, more generally the logarithm of the partition function, , itself receives corrections from the massless spectrum of particles in the theory whose solution contains the BH. These corrections therefore cannot be determined from the BH solution only. They are universal only in the sense that they are independent from the UV completion of the theory (see review article [26]).

Moreover, it must be noted that a given theory of quantum gravity will assign a Hilbert space to counting the number of microstates of the Hilbert space which gives us the entropy of of the BH by Boltzmann’s entropy formulawhere is the Boltzmann constant and is the microstates of only. Analogously, there must exists inner Hilbert space for . Therefore, Boltzmann’s entropy formula for becomeswhere is the microstates of . Finally, their product should beNow we turn to compute the logarithmic corrections to BH entropy product formula by using the CFT formalism.

2. Logarithmic Corrections to the BH Entropy Product Formula of via CFT

We have started with the partition function [11, 24] of any thermodynamic system consisting of which should readwhere can be defined as the temperature of . We have to set Boltzmann constant .

The density of states of the said thermodynamic system may be expressed by taking an inverse Laplace transformation (keeping fixed) of the partition function defined at [24, 27, 28]where is a real constant and definingis the exact entropy of as a function of temperature.

Now if we consider the system to be in equilibrium, then the inverse temperature is defined to be ; therefore, we can expand the entropy function of aswhere and at .

Substituting (9) in (7), we find

Let us put and choosing , is a real variable and evaluating a contour integration, one obtains

Thus, the logarithm of the density of states gives the corrected entropy of :The main aim of this work is to compute the term by using the exact entropy function , evaluated at the equilibrium temperature .

Assume that the exact entropy function of    (the entropy function is defined in [20] for only. We here prescribed that this entropy function is valid for as well) which is followed from the CFT [24] isWe choose the entropy function [24] in more general form (because this form admits a saddle point) aswhere . When we have considered the special case which is dictated by CFT, then .

Now the above function has an extremum value atExpanding around and by evaluating second-order derivative, one obtains whereWe also derived in [11] close to the equilibrium and at the inverse temperature the entropy function of aswhere and at . Comparing (16) and (18), we findInverting these equations, one can find and in terms of and Putting these values in (15), we getNow inverting in terms of and , one obtainsSubstituting these values in (12), we haveThis is in fact the generic formula for leading order correction to Bekenstein-Hawking formula. It should be noted that the formula is indeed independent of and . What is new in this formula is that one could calculate the logarithmic correction to the Bekenstein-Hawking entropy of   without knowing the values of any specific heat of the BH but only knowing the values of    of    and    for the said BH.

Therefore, the product becomes We have already argued the implication of this formula in [11] as when we take the first-order correction, it indicates that the product is always dependent on mass parameter. Therefore, the theorem of Ansorg-Hennig [1], “the area product formula of being independent of mass,” is no longer true when we have taken into consideration the leading order logarithmic correction.

For completeness, we further compute the logarithmic correction of entropy sum, entropy minus, and entropy division using (23). They areThese equations indicate that they are all mass-independent; hence, they are not universal in this sense.

3. Examples

Now we apply this formula for specific BHs in order to calculate the logarithm correction to the Bekenstein-Hawking entropy of . First we take the four-dimensional Reissner-Nordstrøm (RN) BH.

Example 1 (RN BH). The BH entropy and BH temperature [15] becomewhere and and are mass and charge of BH, respectively.
Therefore, the entropy correction is given byWe can conclude that the product includes the mass term so it is not universal.
The second example we take is the Kehagias-Sfetsos (KS) BH [29] in Hoava-Lifshitz gravity [30–32].

Example 2 (KS BH). The entropy for KS BH [8] should readwhere is coupling constant and and are EH and CH, respectively. The Hawking temperature becomesTherefore, the entropy correction for KS BH is given byIt implies that when the logarithmic correction is taken into consideration the entropy product is not mass-independent (universal).
Now we take the AdS space. First, it should be Schwarzschild-AdS space-time (in the limit , one gets the Schwarzschild BH. Here the horizon is at and . Thus, one obtains the logarithmic correction as . Therefore, it indicates that the logarithmic correction must be mass dependent. Therefore, it is not universal).

Example 3 (Schwarzschild-AdS BH). The only physical horizon [4] is atThus, the entropy of should bewhere is cosmological constant. The BH temperature readsNow the entropy correction formula should readIn fact, in both cases, with logarithmic correction and without logarithmic correction, the entropy depends on the mass parameter; thus, it is not universal and therefore it is not quantized.
Now we take the RN-AdS case [4].

Example 4 (RN-AdS BH). The quartic Killing horizon equation becomesThere are at least two real zeros which correspond to two physical horizons, namely, EH, , and CH, .
The entropy should readThe BH temperature of is given byTherefore, the logarithmic correction becomesIt implies that the product of logarithmic correction is not mass-independent.
Now we take the spinning BH.

Example 5 (Kerr BH). The BH entropy and BH temperature [15] arewhere , , and are mass and spin parameter of the BH, respectively.
Now the logarithmic correction is computed to beIt also indicates that whenever we take the logarithmic correction; the entropy product of is not universal.
Next we take charged rotating BH.

Example 6 (Kerr-Newman BH). The BH entropy and BH temperature [15] should readwhere and , , and correspond to the mass, the spin parameter, and the charge of BH, respectively.
The logarithmic correction is derived to beAgain we observe that when we take logarithmic correction the entropy product of for Kerr-Newman BH [1, 6] is not mass-independent.

Example 7 (Kerr-Newman AdS BH). The horizon equation [33] is given bywhich implies that the quartic order horizon equationThis equation has two real zeros which correspond to two physical horizons, namely, , where and are electric and magnetic charge parameters, respectively. The BH entropy and BH temperature areThe logarithmic correction for KN-AdS BH should read

Example 8 (nonrotating BTZ BH). The BH horizon is at . is 3D Newtonian constant. The BH entropy of for BTZ BH isThe BH temperature iswhere is cosmological constant. Thus, the BH entropy correction for BTZ BH isIn fact, it is isolated case and there is only one horizon; therefore, both the logarithmic correction of entropy and without logarithmic corrections term are mass dependent.

Example 9 (rotating BTZ BH). The BH horizons for rotating BTZ BH [34, 35] are given byThe BH entropy of isand the Hawking temperature of should readTherefore, the BH entropy correction is calculated to beIt is clear from the calculation that the product depends on the mass parameter.

Example 10 (charged dilaton BH). The horizons of charged dilation BH [36] are atwhere is given byThe BH temperature of readsThe entropy for both the horizons is Thus, the BH entropy correction for should beAnd the entropy correction for isThis is an interesting case because the entropy product of and the entropy product with logarithmic correction both go to zero value. The logarithmic correction survives for and only when we have taken into account the logarithmic correction for it breaks down and therefore the product also breaks down.

Example 11 (Kerr-Sen BH). The horizon for Kerr-Sen [37, 38] BH is situated atThe BH entropy and BH temperature for Sen BH areTherefore, the logarithmic correction is calculated to be

Example 12 (Sultana-Dyer BH). This is an example of a dynamical cosmological BH [39]. The horizon is located at , where is the mass of BH.
The entropy (the surface area at indicates that the cosmological BH is formed initially from Big-Bang singularity.   is the conformal factor) and temperature of this BH areThe logarithmic correction is found to beThe interesting fact in this case is that we have found that the entropy correction term in the logarithmic correction term is mass-independent, whereas without logarithmic term it is mass-dependent.